Euler Products beyond the Boundary

TL;DR

This paper explores the properties of Euler products related to the Riemann zeta and Dirichlet L-functions on the critical line, introduces the Deep Riemann Hypothesis, and connects zeros to statistical mechanics.

Contribution

It proposes the Deep Riemann Hypothesis as a refined conjecture and offers new interpretations of zeros using concepts from statistical mechanics.

Findings

Analysis of Euler products beyond the boundary

Formulation of the Deep Riemann Hypothesis

Interpretation of zeros via statistical mechanics

Abstract

We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothesis, which is named "the Deep Riemann Hypothesis" (DRH), is examined. We also study various analogs for global function fields. We give an interpretation for the nontrivial zeros from the viewpoint of statistical mechanics.